Simplify the following expression and state the condition under which the simplification is valid. $a = \dfrac{r^2 - 81}{r + 9}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = r$ $ b = \sqrt{81} = 9$ So we can rewrite the expression as: $a = \dfrac{({r} + {9})({r} {-9})} {r + 9} $ We can divide the numerator and denominator by $(r + 9)$ on condition that $r \neq -9$ Therefore $a = r - 9; r \neq -9$